John Carlos Baez on Nostr: Everyone agrees that naive quantum gravity, where you posit a particle called the ...
Everyone agrees that naive quantum gravity, where you posit a particle called the 'graviton' interacting according to rules modeled after general relativity, is nonrenormalizable. But what does this actually mean? How bad is it?
In a renormalizable theory, as you increase the cutoff on the allowed mass of virtual particles, you can keep twiddling the parameters of your theory so that particle masses and charges as measured at low energies stay equal to their observed values.
For a nonrenormalizable theory you have to do more! As you boost the cutoff, you have to keep introducing more and more new interactions and twiddling *their* parameters, too!
In naive quantum gravity it works like this. General relativity only gives 3-graviton and 4-graviton interactions (one is shown below). But from these you can assemble diagrams involving more gravitons. When these diagrams include high-mass virtual gravitons, their effect on low-energy physics is to mimic 5-graviton interactions, 6-graviton interactions, etc. Thus, at low energies we get a theory noticeably different from general relativity! And as the cutoff increases, the problem gets worse: the n-graviton interactions for large n become bigger and bigger at low energies.
To prevent this, we need to *add* 5-graviton interactions, 6-graviton interactions etc. to our theory, with strengths chosen carefully to *cancel out* the problematic effects at low energies. And the higher our cutoff this is, the more of these interactions we need to add, to get a theory that acts like general relativity at low energies.
We might be able to do it. But in the old days nobody wanted to, because it's a huge pain in the ass.
(6/n)
In a renormalizable theory, as you increase the cutoff on the allowed mass of virtual particles, you can keep twiddling the parameters of your theory so that particle masses and charges as measured at low energies stay equal to their observed values.
For a nonrenormalizable theory you have to do more! As you boost the cutoff, you have to keep introducing more and more new interactions and twiddling *their* parameters, too!
In naive quantum gravity it works like this. General relativity only gives 3-graviton and 4-graviton interactions (one is shown below). But from these you can assemble diagrams involving more gravitons. When these diagrams include high-mass virtual gravitons, their effect on low-energy physics is to mimic 5-graviton interactions, 6-graviton interactions, etc. Thus, at low energies we get a theory noticeably different from general relativity! And as the cutoff increases, the problem gets worse: the n-graviton interactions for large n become bigger and bigger at low energies.
To prevent this, we need to *add* 5-graviton interactions, 6-graviton interactions etc. to our theory, with strengths chosen carefully to *cancel out* the problematic effects at low energies. And the higher our cutoff this is, the more of these interactions we need to add, to get a theory that acts like general relativity at low energies.
We might be able to do it. But in the old days nobody wanted to, because it's a huge pain in the ass.
(6/n)